International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 251-252
Section 4.2.6.3.3.1. Measurements in the high-energy limit
D. C. Creaghb
|
In this case, there is some possibility of testing the validity of the relativistic dipole and relativistic multipole theories since, in the high-energy limit, the value of must approach a value related to the total self energy of the atom
. That there is an atomic number dependent systematic error in the relativistic dipole approach has been demonstrated by Creagh (1984
). The question of whether the relativistic multipole approach yields a result in better accord with the experimental data is answered in Table 4.2.6.4
, where a comparison of values of
is made for three theoretical data sets (this work; Cromer & Liberman, 1981
; Wagenfeld, 1975
) with a number of experimental results. These include the `direct' measurements using X-ray interferometers (Cusatis & Hart, 1975
; Creagh, 1984
), the Kramers–Kronig integration of X-ray attenuation data (Gerward et al., 1979
), and the angle-of-the-prism data of Deutsch & Hart (1984b
). Also included in the table are `indirect' measurements: those of Price et al. (1978
), based on Pendellösung measurements, and those of Grimvall & Persson (1969
). These latter data estimate
and not
. Table 4.2.6.4
details values of the real part of the dispersion correction for LiF, Si, Al and Ge for the characteristic wavelengths Ag
, Mo
and Cu
. Of the atomic species listed, the first three are approaching the high-energy limit at Ag
, whilst for germanium the K-shell absorption edge lies between Mo
and Ag
.
|
The high-energy-limit case is considered first: both the relativistic dipole and relativistic multipole theories underestimate for LiF whereas the non-relativistic theory overestimates
when compared with the experimental data. For silicon, however, the relativistic multipole yields values in good agreement with experiment. Further, the values derived from the work of Takama et al. (1982
), who used a Pendellösung technique to measure the atomic form factor of aluminium are in reasonable agreement with the relativistic multipole approach. Also, some relatively imprecise measurements by Creagh (1985
) are in better accordance with the relativistic multipole values than with the relativistic dipole values.
Further from the high-energy limit (smaller values of , the relativistic multipole approach appears to give better agreement with theory. It must be reported here that measurements by Katoh et al. (1985a
) for lithium fluoride at a wavelength of 0.77366 Å yielded a value of 0.018 in good agreement with the relativistic multipole value 0.017.
At still smaller values of , the non-relativistic theory yields values considerably at variance with the experimental data, except for the case of LiF using Cu
radiation. The relativistic multipole approach seems, in general, to be a little better than the relativistic approach, although agreement between experiment and theory is not at all good for germanium. Neither of the experiments cited here, however, has claims to high accuracy.
In Table 4.2.6.5, a comparison is made of measurements of
derived from the results of the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987
, 1990
) with a number of theoretical predictions. The measurements were made on carbon, silicon and copper specimens at the characteristic wavelengths Cu
, Mo
and Ag
. The principal conclusion that can be drawn from perusal of Table 4.2.6.5
is that only minor, non-systematic differences exist between the predictions of the several relativistic approaches and the experimental results. In contrast, the non-relativistic theory fails for higher values of atomic number.
|
References















